PARAMETRIC STATISTICAL TESTS
testing the equality of
EXPECTED VALUES of 2 RV’s
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED
the usual case:
Given RV X follows — in two (different) populations distributions
N (µ1 , σ1 ),
N (µ2 , σ2 ); σ1 = σ2 = σ (unknown)
(these do not have to be The Normal distributions!)
HYPOTHESIS TO BE VERIFIED: H0 : µ1 = µ2 ≡ µ
We form two samples (each from every population):
sample 1: X1 , X2 , . . ., Xn1 size n1
sample 2: Y1 , Y2 , . . ., Yn2 size n2
and we construct the ”usual” estimators:
X̄ =
n1
n2
1 X
1 X
Xk Ȳ =
Yk
n1
n2
k=1
and
2
SX
=
k=1
n1
n2
1 X
1 X
(Xk − X̄)2 SY2 =
(Yk − Ȳ )2
n1
n2
k=1
k=1
the two average values X̄ and Ȳ have normal distributions:
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED
the two average values X̄ and Ȳ have normal distributions:
X̄ =
ˆ N
σ
µ, √
n1
Ȳ =
ˆ N
σ
µ, √
n2
so their difference has the distribution:
r
n1 + n2
X̄ − Ȳ =N
ˆ
0, σ
n1 n2
as we have
σ 2 (X̄ − Ȳ ) = σ 2 (X̄) + σ 2 (Ȳ ) =
σ2
n1 + n2
σ2
+
= σ2
n1
n2
n1 n2
Now — we have to transform the normal distribution into the Student’s
distribution (we don’t know σ!) — following the same scheme we
employed in the case of the confidence intervals:
N=
ˆ U≡
X̄ − µ √
X̄ − µ √
n→
n−1≡t=
ˆ St
σ
S
this transformation consists of two steps:
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED
this transformation consists of two steps:
N=
ˆ U≡
1
X̄ − µ √
X̄ − µ √
n→
n−1≡t=
ˆ St
σ
S
we divide U by the RV
nS 2
σ2
2
√
nS/σ, whose square:
follows the distribution χ2n−1
we multiply the resulting
√ quantity by the square–root of the number
of degrees of freedom n − 1
For the case of 2 independent samples the variable analogous to
2
n1 SX
+ n2 SY2
and the number of degrees of freedom is
nS 2 /σ 2 is:
σ2
equal to: n1 − 1 + n2 − 1.
If so, the standardised variable
must be divided by
p
X̄ − Ȳ
r
n1 + n2
σ
n1 n2
2 + n S 2 /σ and multiplied by
n1 SX
2 Y
√
n1 + n2 − 2.
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED
the standardised variable:
X̄ − Ȳ
r
n1 + n2
σ
n1 n2
√
2 + n S 2 /σ and multiplied by
n1 SX
n1 + n2 − 2.
2 Y
s
X̄ − Ȳ
n1 n2 (n1 + n2 − 2)
∆ = p
×
2
2
n1 + n2
n1 SX + n2 SY
must be divided by
p
follows Student’s distribution with n1 + n2 − 2 degrees of freedom.
Testing H0 : µ1 = µ2 consists in verifying whether the value of
∆(calculated from the 2 samples) ∈ Ucr (α).
The critical (rejection) region Ucr (α, H1 ) is defined by appropriate
quantiles of the Student distribution.
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED
Testing H0 : µ1 = µ2 – critical (rejection) region
Ucr (α, H1 ) may be: (nn = n1 + n2 − 2)
1
H1 : µ1 < µ2 ; Ucr = ( − ∞, −t(1 − α, nn)]
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED
Testing H0 : µ1 = µ2 – critical (rejection) region
Ucr (α, H1 ) may be: (nn = n1 + n2 − 2)
2
H1 : µ1 > µ2 ; Ucr = [ + t(1 − α, nn), ∞)
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED
Testing H0 : µ1 = µ2 – critical (rejection) region
Ucr (α, H1 ) may be: (nn = n1 + n2 − 2)
3
H1 µ1 6= µ2 ;
Ucr = ( − ∞, −t(1 − α2 , nn) )
S
( + t(1 − α2 , nn), ∞)
PARAMETRIC STATISTICAL TESTS testing the equality of EXPECTED